Links of singular points of complex hypersurfaces

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Contents

1 Introduction

The links of singular points of complex hypersurfaces provide a large class of examples of highly-connected odd dimensional manifolds. A standard reference is [Milnor1968]. See also [Hirzebruch&Mayer1968] and [Dimca1992].

These manifolds are the boundaries of highly-connected, stably parallelisable even dimensional manifolds and hence stably parallelisable themselves. In the case of singular points of complex curves, the link of such a singular point is a fibered link in the 3-dimensional sphere S^3.

2 Construction and properties

Let f(z_1, \dots, z_{n+1}) be a non-constant polynomial in n+1 complex variables. A complex hypersurface V is the algebraic set consisting of points z=(z_1, \dots, z_{n+1}) such that f(z)=0. A regular point z \in V is a point at which some partial derivative \partial f /\partial z_j does not vanish; if at a point z \in V all the partial derivatives \partial f / \partial z_j vanish, z is called a singular point of V.

Near a regular point z, the complex hypersurface V is a smooth manifold of real dimension 2n; in a small neighborhood of a singular point z, the topology of the complex hypersurface V is more complicated. One way to look at the topology near z, due to Brauner, is to look at the intersection of V with a (2n+1)-dimensioanl sphere of small radius \epsilon S_{\epsilon} centered at z.

\begin{thm} The space K=V\cap S_{\epsilon} is (n-2)-connected. \end{thm}

The homeomorphism type of K is independent of the small parameter \epsilon, it is called the link of the singular point z.

\begin{thm}(Fibration Theorem) For \epsilon sufficiently small, the space S_{\epsilon}-K is a smooth fiber bundle over S^1, with projection map \phi \colon S_{\epsilon}-K \to S^1, z \mapsto f(z)/|f(z)|. Each fiber F_{\theta} is parallelizable and has the homotopy type of a finite CW-complex of dimension n. \end{thm}

The fiber F_{\theta} is usually called the Milnor fiber of the singular point z.

A singular point z is isolated if there is no other singular point in some small neighborhood of z.

In this special situation, the above theorems are strengthened to the following

\begin{thm} Each fiber F_{\theta} is a smooth parallelizable manifold, the closure \overline{F_{\theta}} has boundary K and the homotopy type of a bouquet of n-spheres S^n\vee \cdots \vee S^n. \end{thm}

3 Invariants

Seen from the above section, the link K of an isolated singular point z of a complex hypersurface V of complex dimension n is an (n-2)-connected (2n-1)-dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a (2n-1)-dimensional closed manifold M which is not a homotopy sphere, (n-2) is the highest connectivity M could have. Therefore to understand the classification and invariants of the links K one needs to understand the classification and invariants of highly-connected odd dimensional manifolds.

On the other hand, as the link K is closely related to the singular point z of the complex hypersurfaces, some of the topological invariants of K are computable from the polynomial.

Let V be a complex hypersurface defined by f(z_1, \dots, z_{n+1}), z^0 be an isolated singular point of V. Let g_j=\partial f /\partial z_j j=1, \dots, n+1. By putting all these g_j's together we get the gradient field of f, which can be viewed as a map g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}, z \mapsto (g_1(z), \dots, g_{n+1}(z)). If z^0 is an isolated singular point, then z \mapsto g(z)/||g(z)|| is a well-defined map from a small sphere S_{\epsilon} centered at z^0 to the unit sphere S^{2n+1} of \mathbb C^{n+1}. The mapping degree \mu is called the multiplicity of the isolated singular point z^0. (\mu is also called the Milnor number of z^0.)

\begin{thm} The middle homology group H_n(F_{\theta}) is a free abelian group of rank \mu. \end{thm}

Furthermore, the homology groups of the link K are determined from the long homology exact sequence

\displaystyle  \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0

of the pair (\overline{F_{\theta}},K). The map j_* is the adjoint of the intersection pairing on \overline{F_{\theta}}

\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.

A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. s is called the Milnor lattice of the singular point. Thus the homology groups of the link K is completely determined by the Milnor lattice of the singular point.

Topological spheres as links of singular points

Especially, the link K is an integral homology sphere if and only if the intersection form s is unimodular, i.~e.~the matrix of s has determinant \pm1. If n\ge 3, the Generalized Poincare Conjecture implies that K is a topological sphere.

By Theorem 2, there is a smooth fiber bundle over S^1 with fiber F_{\theta}. The natural action of a generator of \pi_1(S^1) induces the characteristic homeomorphism h of the fiber F_0=\phi^{-1}. Let h_* \colon H_n(F_0) \to H_n(F_0) be the induced isomorphism on homology and \Delta(t)=\det(tI-h_*) be the characteristic polynomial of the linear transformation h_*. It's a consequence of the Wang sequence associated with the fiber bundle over S^1 that

Lemma 5.1. For n \ne 2 the manifolds K is a topological sphere is and only if the integer \Delta(1)=\det(I-h_*) equals to \pm 1.

When K is a topological sphere, as it is the boundary of an (n-1)-connected parallelisable 2n-manifold \overline{F_0}, our knowledge of exotic spheres allows us to determine the diffeomorphism class of K completely:

  • if n is even, the diffeomorphism class of K is determined by the signature of the intersection pairing
\displaystyle s \colon H_n(\overline{F_0}) \otimes  H_n(\overline{F_0}) \to \mathbb Z
\displaystyle c(\overline{F_0}) \in \mathbb Z_2

which was computed in [Levine1966]

\displaystyle c(\overline{F_0})=0  \ \ \textrm{if}  \ \  \Delta(-1) \equiv \pm 1 \pmod 8
\displaystyle c(\overline{F_0})=1  \ \ \textrm{if}  \ \  \Delta(-1) \equiv \pm 3 \pmod 8

4 Examples

The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form

\displaystyle f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}

where a_1, \dots, a_{n+1} are integers \ge 2. The origin is an isolated singular point of f.

\begin{thm}(Brieskorn-Pham) The group H_n(F_{\theta}) is free abelian of rank

\displaystyle \mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)

The characteristic polynomial is

\displaystyle \Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),

where each \omega_j ranges over all a_j-th root of unit other than 1. \end{thm}

The link K is called a Brieskorn variety.

For a_1=\cdots=a_{n+1}=2, it's seen from the defining equations that the link K is the sphere bundle of the tangent bundle of the n-sphere, i.~e.~the Stiefel manifold V_{n+1,2}(\mathbb R).

The simplest nontrivial example is a_1=\cdots =a_n=2, a_{n+1}=3. Then \omega_1 = \cdots \omega_n=-1, \omega_{n+1}=(-1\pm \sqrt{3})/2. The characteristic polynomial is

\displaystyle \Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}
\displaystyle \Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.

For n=2k+1 we have \Delta(1)=1 so the link K is a topological sphere of dimension 4k+1; \Delta(-1)=3, thus by [Levine1966] K has nontrivial Kervaire invariant. Especially for k=2 K^9 is the Kervaire sphere.

The above example is a special case of the A_k-singularities, whose defining polynomial is

\displaystyle f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}

k being an integer \ge 1. The Milnor lattice of an A_k-singularity is represented by the Dynkin diagram of the simple Lie algebra A_k. When n=3, the diffeomorphism classification of the link K is obtained from its Milnor lattice and the classification of simply-connected 5-manifolds:

  • K is diffeomorphic to S^2 \times S^3 if k is odd;
  • K is diffeomorphic to S^5 if k is even.

In this dimension, the diffeomorphism classification of the link K of other types of singular points can be obtained in the same way, once we know the Milnor lattice of the singular point.

5 Further discussion

The link K of a singular point z is the intersection of the hypersurface V defined by f and the sphere S_{\epsilon} in the ambient space \mathbb C^{n+1}. Therefore it inherits certain symmetries from that of the singular point. As an example, consider an A_k-singularity in \mathbb C^4. There is an orientation preserving involution

\displaystyle \tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, _3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).

\tau_k induces an orientation preserving free involution of K\cong S^5 or S^2 \times S^3. For k=0,2,4,6, \tau_k's provide all the 4 smooth free involutions on S^5 (see [Geiges&Thomas1998]).

6 References

$. \end{thm} The link $K$ is called a [[Exotic spheres#Brieskorn varieties|Brieskorn variety]]. For $a_1=\cdots=a_{n+1}=2$, it's seen from the defining equations that the link $K$ is the sphere bundle of the tangent bundle of the $n$-sphere, i.~e.~the [[wikipedia:Stiefel manifold|Stiefel manifold]] $V_{n+1,2}(\mathbb R)$. The simplest nontrivial example is $a_1=\cdots =a_n=2$, $a_{n+1}=3$. Then $\omega_1 = \cdots \omega_n=-1$, $\omega_{n+1}=(-1\pm \sqrt{3})/2$. The characteristic polynomial is $$\Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}$$ $$\Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.$$ For $n=2k+1$ we have $\Delta(1)=1$ so the link $K$ is a topological sphere of dimension k+1$; $\Delta(-1)=3$, thus by {{cite|Levine1966}} $K$ has nontrivial [[wikipedia:Kervaire invariant|Kervaire invariant]]. Especially for $k=2$ $K^9$ is the [[Exotic spheres#Plumbing|Kervaire sphere]]. The above example is a special case of the $A_k$-singularities, whose defining polynomial is $$f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$$ $k$ being an integer $\ge 1$. The Milnor lattice of an $A_k$-singularity is represented by the Dynkin diagram of the simple Lie algebra $A_k$. When $n=3$, the diffeomorphism classification of the link $K$ is obtained from its Milnor lattice and the classification of [[5-manifolds: 1-connected|simply-connected 5-manifolds]]: * $K$ is diffeomorphic to $S^2 \times S^3$ if $k$ is odd; * $K$ is diffeomorphic to $S^5$ if $k$ is even. In this dimension, the diffeomorphism classification of the link $K$ of other types of singular points can be obtained in the same way, once we know the [[Links of singular points of complex hypersurfaces#Invariants|Milnor lattice]] of the singular point. == Further discussion == ; The link $K$ of a singular point $z$ is the intersection of the hypersurface $V$ defined by $f$ and the sphere $S_{\epsilon}$ in the ambient space $\mathbb C^{n+1}$. Therefore it inherits certain symmetries from that of the singular point. As an example, consider an $A_k$-singularity in $\mathbb C^4$. There is an orientation preserving involution $$\tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, _3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).$$ $\tau_k$ induces an orientation preserving free involution of $K\cong S^5$ or $S^2 \times S^3$. For $k=0,2,4,6$, $\tau_k$'s provide all the 4 smooth free involutions on $S^5$ (see {{cite|Geiges&Thomas1998}}). == References == {{#RefList:}} [[Category:Manifolds]]3-dimensional sphere S^3.

2 Construction and properties

Let f(z_1, \dots, z_{n+1}) be a non-constant polynomial in n+1 complex variables. A complex hypersurface V is the algebraic set consisting of points z=(z_1, \dots, z_{n+1}) such that f(z)=0. A regular point z \in V is a point at which some partial derivative \partial f /\partial z_j does not vanish; if at a point z \in V all the partial derivatives \partial f / \partial z_j vanish, z is called a singular point of V.

Near a regular point z, the complex hypersurface V is a smooth manifold of real dimension 2n; in a small neighborhood of a singular point z, the topology of the complex hypersurface V is more complicated. One way to look at the topology near z, due to Brauner, is to look at the intersection of V with a (2n+1)-dimensioanl sphere of small radius \epsilon S_{\epsilon} centered at z.

\begin{thm} The space K=V\cap S_{\epsilon} is (n-2)-connected. \end{thm}

The homeomorphism type of K is independent of the small parameter \epsilon, it is called the link of the singular point z.

\begin{thm}(Fibration Theorem) For \epsilon sufficiently small, the space S_{\epsilon}-K is a smooth fiber bundle over S^1, with projection map \phi \colon S_{\epsilon}-K \to S^1, z \mapsto f(z)/|f(z)|. Each fiber F_{\theta} is parallelizable and has the homotopy type of a finite CW-complex of dimension n. \end{thm}

The fiber F_{\theta} is usually called the Milnor fiber of the singular point z.

A singular point z is isolated if there is no other singular point in some small neighborhood of z.

In this special situation, the above theorems are strengthened to the following

\begin{thm} Each fiber F_{\theta} is a smooth parallelizable manifold, the closure \overline{F_{\theta}} has boundary K and the homotopy type of a bouquet of n-spheres S^n\vee \cdots \vee S^n. \end{thm}

3 Invariants

Seen from the above section, the link K of an isolated singular point z of a complex hypersurface V of complex dimension n is an (n-2)-connected (2n-1)-dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a (2n-1)-dimensional closed manifold M which is not a homotopy sphere, (n-2) is the highest connectivity M could have. Therefore to understand the classification and invariants of the links K one needs to understand the classification and invariants of highly-connected odd dimensional manifolds.

On the other hand, as the link K is closely related to the singular point z of the complex hypersurfaces, some of the topological invariants of K are computable from the polynomial.

Let V be a complex hypersurface defined by f(z_1, \dots, z_{n+1}), z^0 be an isolated singular point of V. Let g_j=\partial f /\partial z_j j=1, \dots, n+1. By putting all these g_j's together we get the gradient field of f, which can be viewed as a map g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}, z \mapsto (g_1(z), \dots, g_{n+1}(z)). If z^0 is an isolated singular point, then z \mapsto g(z)/||g(z)|| is a well-defined map from a small sphere S_{\epsilon} centered at z^0 to the unit sphere S^{2n+1} of \mathbb C^{n+1}. The mapping degree \mu is called the multiplicity of the isolated singular point z^0. (\mu is also called the Milnor number of z^0.)

\begin{thm} The middle homology group H_n(F_{\theta}) is a free abelian group of rank \mu. \end{thm}

Furthermore, the homology groups of the link K are determined from the long homology exact sequence

\displaystyle  \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0

of the pair (\overline{F_{\theta}},K). The map j_* is the adjoint of the intersection pairing on \overline{F_{\theta}}

\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.

A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. s is called the Milnor lattice of the singular point. Thus the homology groups of the link K is completely determined by the Milnor lattice of the singular point.

Topological spheres as links of singular points

Especially, the link K is an integral homology sphere if and only if the intersection form s is unimodular, i.~e.~the matrix of s has determinant \pm1. If n\ge 3, the Generalized Poincare Conjecture implies that K is a topological sphere.

By Theorem 2, there is a smooth fiber bundle over S^1 with fiber F_{\theta}. The natural action of a generator of \pi_1(S^1) induces the characteristic homeomorphism h of the fiber F_0=\phi^{-1}. Let h_* \colon H_n(F_0) \to H_n(F_0) be the induced isomorphism on homology and \Delta(t)=\det(tI-h_*) be the characteristic polynomial of the linear transformation h_*. It's a consequence of the Wang sequence associated with the fiber bundle over S^1 that

Lemma 5.1. For n \ne 2 the manifolds K is a topological sphere is and only if the integer \Delta(1)=\det(I-h_*) equals to \pm 1.

When K is a topological sphere, as it is the boundary of an (n-1)-connected parallelisable 2n-manifold \overline{F_0}, our knowledge of exotic spheres allows us to determine the diffeomorphism class of K completely:

  • if n is even, the diffeomorphism class of K is determined by the signature of the intersection pairing
\displaystyle s \colon H_n(\overline{F_0}) \otimes  H_n(\overline{F_0}) \to \mathbb Z
\displaystyle c(\overline{F_0}) \in \mathbb Z_2

which was computed in [Levine1966]

\displaystyle c(\overline{F_0})=0  \ \ \textrm{if}  \ \  \Delta(-1) \equiv \pm 1 \pmod 8
\displaystyle c(\overline{F_0})=1  \ \ \textrm{if}  \ \  \Delta(-1) \equiv \pm 3 \pmod 8

4 Examples

The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form

\displaystyle f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}

where a_1, \dots, a_{n+1} are integers \ge 2. The origin is an isolated singular point of f.

\begin{thm}(Brieskorn-Pham) The group H_n(F_{\theta}) is free abelian of rank

\displaystyle \mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)

The characteristic polynomial is

\displaystyle \Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),

where each \omega_j ranges over all a_j-th root of unit other than 1. \end{thm}

The link K is called a Brieskorn variety.

For a_1=\cdots=a_{n+1}=2, it's seen from the defining equations that the link K is the sphere bundle of the tangent bundle of the n-sphere, i.~e.~the Stiefel manifold V_{n+1,2}(\mathbb R).

The simplest nontrivial example is a_1=\cdots =a_n=2, a_{n+1}=3. Then \omega_1 = \cdots \omega_n=-1, \omega_{n+1}=(-1\pm \sqrt{3})/2. The characteristic polynomial is

\displaystyle \Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}
\displaystyle \Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.

For n=2k+1 we have \Delta(1)=1 so the link K is a topological sphere of dimension 4k+1; \Delta(-1)=3, thus by [Levine1966] K has nontrivial Kervaire invariant. Especially for k=2 K^9 is the Kervaire sphere.

The above example is a special case of the A_k-singularities, whose defining polynomial is

\displaystyle f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}

k being an integer \ge 1. The Milnor lattice of an A_k-singularity is represented by the Dynkin diagram of the simple Lie algebra A_k. When n=3, the diffeomorphism classification of the link K is obtained from its Milnor lattice and the classification of simply-connected 5-manifolds:

  • K is diffeomorphic to S^2 \times S^3 if k is odd;
  • K is diffeomorphic to S^5 if k is even.

In this dimension, the diffeomorphism classification of the link K of other types of singular points can be obtained in the same way, once we know the Milnor lattice of the singular point.

5 Further discussion

The link K of a singular point z is the intersection of the hypersurface V defined by f and the sphere S_{\epsilon} in the ambient space \mathbb C^{n+1}. Therefore it inherits certain symmetries from that of the singular point. As an example, consider an A_k-singularity in \mathbb C^4. There is an orientation preserving involution

\displaystyle \tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, _3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).

\tau_k induces an orientation preserving free involution of K\cong S^5 or S^2 \times S^3. For k=0,2,4,6, \tau_k's provide all the 4 smooth free involutions on S^5 (see [Geiges&Thomas1998]).

6 References

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